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\begin{document}

\title{高等代数二}
\subtitle{7-2-线性变换的运算 }
%\institute{上海立信会计金融学院}
%\author{王立庆}
\author{{\ppr LQW}}
\renewcommand{\today}{{\ppr \number\year \,年 \number\month \,月 \number\day \,日} }
%\date{{\ppr 2023年3月9日} }

\maketitle

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%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{7.2.i. 作业：星期天晚上十点半之前在网络教学平台提交 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
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\begin{enumerate}
\item   整理课堂笔记，补充没写完的计算或证明。
\item   习题(7.2)\#2,4,6, 抄写题目。
\end{enumerate}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.2.ii. 目录 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
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\begin{enumerate}
\item  线性变换的定义
\item  两个线性变换的和
\item  零变换和负变换
\item  线性变换的数乘
\item  定理7.2.1. 线性变换全体组成的向量空间
\item  线性变换的幂次
\item  线性变换的多项式
\item  线性变换的逆

\end{enumerate}

\end{frame}

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%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{7.2.iii. 课堂讲解重点 }

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\begin{enumerate}

\item  线性变换的概念和例子
\item  线性变换全体组成的向量空间
%\item  线性变换的多项式
%\item  线性变换的逆

\end{enumerate}

\end{frame}


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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.2.1. 线性变换的定义}

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\begin{itemize}

\item  {\color{red}问题：什么是线性变换？}

\vspace{0.3cm}

\item 解答：向量空间到自身的线性映射就称为线性变换。即一个变换 
$$\sigma: V\to V,$$ 
满足下述条件：对任意 $\alpha_1,\alpha_2\in V$ 和任意 $k_1,k_2\in F$, 都有 
\begin{eqnarray*}
\sigma(k_1\alpha_1+k_2\alpha_2) = k_1\sigma(\alpha_1)+k_2\sigma(\alpha_2).
\end{eqnarray*}

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.2.2.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  {\color{red}问题：符号 $L(V)$ 是什么含义？}

\vspace{0.3cm}

\item 解答：符号 $L(V)$ 是向量空间 $V$ 到自身的所有线性变换全体组成的集合。例如：$\sigma\in L(V)$ 表示 $\sigma:V\to V$ 是一个线性变换。

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.2.3. 两个线性变换的和}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}问题：设 $\sigma,\tau:V\to V$ 是两个线性变换，则 $\sigma+\tau$ 指的是什么？} 

\vspace{0.3cm}

\item 解答：符号 $\sigma+\tau$ 也是向量空间 $V$ 到自身的一个变换，定义为
\begin{eqnarray*}
(\sigma+\tau)(\alpha) = \sigma(\alpha) + \tau(\alpha).
\end{eqnarray*}
换种写法，也可以写成
\begin{eqnarray*}
\sigma+\tau: V &\to& V \\
\alpha &\mapsto& \sigma(\alpha) + \tau(\alpha).
\end{eqnarray*}

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.2.4.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  {\color{red}求证：设 $\sigma,\tau:V\to V$ 是两个线性变换， $\sigma+\tau$ 也是一个线性变换。 } 

\vspace{0.3cm}

\item  证明：
\begin{eqnarray*}
{\color{red}(\sigma+\tau)}(k_1\alpha_1+k_2\alpha_2) &=& \sigma(k_1\alpha_1+k_2\alpha_2) + \tau(k_1\alpha_1+k_2\alpha_2) \\
&=& k_1\sigma(\alpha_1)+k_2\sigma(\alpha_2) + k_1\tau(\alpha_1)+k_2\tau(\alpha_2) \\
&=& k_1{\color{red}(\sigma+\tau)}(\alpha_1) + k_2{\color{red}(\sigma+\tau)}(\alpha_2).
\end{eqnarray*}
 
 \item  其中：
 \begin{enumerate}
 \item  第一个等号是根据 $\sigma+\tau$ 的定义。
 \item  第二个等号是根据 $\sigma$ 和 $\tau$ 都是线性变换。
 \item  第三个等号是向量空间中的加法和数乘的性质，以及 $\sigma+\tau$ 的定义。
 \end{enumerate}

\end{itemize}

\end{frame}


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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.2.5. 零变换和负变换}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}问题：什么是零变换？什么是线性变换 $\sigma:V\to V$ 的负变换？} 

\vspace{0.3cm}

\item 解答：
\begin{enumerate}
\item 向量空间 $V$ 上的{\color{red}零变换}，就是将所有向量都映到零向量的变换，即下述变换，其中 $\theta_V$ 是 $V$ 中的零向量，
\begin{eqnarray*}
{\color{red}\theta}: V &\to& V \\
\alpha &\mapsto& \theta_V.
\end{eqnarray*}

\item 线性变换 $\sigma:V\to V$ 的{\color{red}负变换}的定义如下，即 $(-\sigma)(\alpha) = -\sigma(\alpha)$,   
\begin{eqnarray*}
{\color{red}-\sigma}: V &\to& V \\
\alpha &\mapsto& -\sigma(\alpha).
\end{eqnarray*}

\end{enumerate}

\end{itemize}

\end{frame}


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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.2.6. 线性变换的数乘}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}问题：设 $\sigma:V\to V$ 是一个线性变换，设 $k\in F$, 则 $k\sigma$ 指的是什么？}

\vspace{0.3cm}

\item 解答：符号 $k\sigma$ 是指这样一个变换
\begin{eqnarray*}
{\color{red}k\sigma}: V &\to& V \\
\alpha &\mapsto& k\sigma(\alpha).
\end{eqnarray*}
也就是说，${\color{red}(k\sigma)}(\alpha) = k\sigma(\alpha)$. 
%\item 

\end{itemize}

\end{frame}


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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.2.7. 定理7.2.1. }

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%每页详细内容

\begin{itemize}
\item  {\color{red}定理：数域 $F$ 上的向量空间 $V$ 到自身的线性变换全体，在上述定义的加法和数乘运算之下，组成 $F$ 上的一个向量空间。}

\vspace{0.3cm}

\item 证明：记向量空间 $V$ 上的所有线性变换全体组成的集合为 $L(V)$.
\begin{enumerate}
\item 首先验证两个线性变换的和，以及一个数乘以一个线性变换，结果仍然是线性变换。设 $\sigma,\tau\in L(V)$, 设 $k\in F$, 要证明 $\sigma+\tau\in L(V)$ 以及 $k\sigma\in L(V)$. 

\item 然后验证八条运算规律。特别要指出这个向量空间中的零向量和负向量。
\begin{enumerate}
\item[(1-2)] 验证 $\sigma+\tau = \tau+\sigma$ 和 $(\sigma+\tau) +\xi = \sigma+ (\tau+\xi)$. 
\item[(3-4)] 零映射 $\theta$ 符合零向量的要求，$\sigma$ 的负向量是 $-\sigma$, 如前所述。
\item[(5)] 验证 $k(\sigma+\tau) = k\sigma+k\tau$.
\item[(6)] 验证 $(k+\ell)\sigma = k\sigma + \ell \sigma$. 
\item[(7)] 验证 $(k(\ell\sigma) = (k\ell) \sigma$.
\item[(8)] 验证 $1\sigma = \sigma$. 具体：$\forall \alpha\in V:\, (1\sigma)(\alpha)=1\sigma(\alpha)=\sigma(\alpha)$.
\end{enumerate}

\end{enumerate}

\end{itemize}

\end{frame}


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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.2.8. 线性变换的幂次}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}问题：设 $\sigma:V\to V$ 是一个线性变换，设 $n$ 是一个非负整数，则 $\sigma^n$ 指的是什么？}

\vspace{0.3cm}

\item 解答：
\begin{itemize}
\item 若 $n=0$, 则 $\sigma^0$ 定义为恒等映射，即 $\iota:\, V\to V$, $\forall \alpha\in V:\, \iota(\alpha) = \alpha$.  
\item 若 $n=1$, 则 $\sigma^1$ 就是 $\sigma$ 本身。
\item 若 $n=2$, 则 $\sigma^2$ 定义为 $\sigma\circ\sigma$, 即 $\sigma^2:V\to V$, $\forall \alpha\in V:\, \sigma^2(\alpha) = \sigma(\sigma(\alpha))$. 
\item 若 $n=3$, 则 $\sigma^3$ 定义为 $\sigma\circ\sigma\circ\sigma$, 即 
\begin{eqnarray*}
\sigma^3:V &\to& V, \\
 \alpha &\mapsto & \sigma(\sigma(\sigma(\alpha))). 
\end{eqnarray*}
\item 若 $n$ 是负整数，就要先假设 $\sigma$ 的逆映射存在。
\end{itemize}

%\item 

\end{itemize}

\end{frame}


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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.2.9. 线性变换的多项式}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}问题：设 $\sigma:V\to V$ 是一个线性变换，设 $f(x)$ 是一个多项式，则 $f\,(\sigma)$ 指的是什么？举例说明。}

\vspace{0.3cm}

\item 解答：符号 $f\,(\sigma)$ 指的仍是一个线性变换 $f\,(\sigma):V\to V$.

\item 例子：设 $f(x)=x^2+3x+2$, 则 $f(\sigma)$ 是这样的一个线性变换：
\begin{eqnarray*}
f\,(\sigma):V &\to& V, \\
 \alpha &\mapsto & \sigma(\sigma(\alpha)) + 3\sigma(\alpha) + 2\alpha. 
\end{eqnarray*}

\item  注：有一个神奇的定理：

设 $V$ 是一个 $n$ 维的向量空间，设 $\sigma$ 是 $V$ 上的一个线性变换。则一定存在一个次数不超过 $n$ 的多项式 $f\,(x)$, 使得 $f\,(\sigma)$ 是零变换。


\end{itemize}

\end{frame}


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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{7.2.10. 线性变换的逆}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}问题：设 $\sigma:V\to V$ 是一个线性变换，什么是它的逆变换？什么时候称一个线性变换是可逆的或者非奇异的？}

\vspace{0.3cm}

\item 解答：当这个线性变换是一个双射的时候，它有逆变换。即这个线性变换既是单射又是满射。可用下述符号表示，
\begin{eqnarray*}
\sigma:V &\rightarrow & V \\
V &\leftarrow & V : \sigma^{-1} 
\end{eqnarray*}

\item 当一个线性变换是双射的时候，称它是可逆的或者非奇异的。

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{6.1.1. }
\begin{frame}{习题(7.2)\#1 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}问题：举例说明，线性变换的乘法不满足交换律。} 

\vspace{0.3cm}

\item 解答：设 $V=\mathbb{R}[x]$ 是一元实系数多项式全体组成的向量空间。设有两个变换分别定义如下，则有 $\sigma\tau - \tau\sigma = \iota$, 
{\small 
\begin{eqnarray*}
\sigma: V\to V, && f\,(x)\mapsto f\,'(x), \\
\tau: V\to V, && f\,(x)\mapsto xf\,(x).
\end{eqnarray*}
}
验证如下。对任意 $f(x)\in V$, 都有 
{\small 
\begin{eqnarray*}
(\sigma\tau - \tau\sigma)(f\,(x)) &=& \sigma(\tau(f\,(x))) - \tau(\sigma(f\,(x))) \\
&=& \sigma(xf\,(x)) - \tau(f\,'(x)) \\
&=& (xf\,(x))\,' - xf\,'(x) \\
&=& 1f\,(x)+xf\,'(x) - xf\,'(x) \\
&=& f\,(x) = \iota(f\,(x)). 
\end{eqnarray*}
}

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{习题(7.2)\#2 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  {\color{red}问题：在 $F[x]$ 中，定义 $\sigma(f(x))=f\,'(x)$, 以及 $\tau(f(x))=xf(x)$. 证明 $\sigma$ 和 $\tau$ 都是 $F[x]$ 到自身的线性变换，并且对任意正整数 $n$ 都有 $$\sigma^n\tau-\tau\sigma^n=n\sigma^{n-1}. $$  }

%\vspace{0.3cm}

\item  思路：按照线性变换的定义验证。按照复合映射的定义计算。


\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{习题(7.2)\#4 }

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%每页详细内容

\begin{itemize}


\item  {\color{red}问题：设 $\sigma\in L(V)$, $\xi\in V$. 设 $\xi, \sigma(\xi), \cdots, \sigma^{k-1}(\xi)$ 都不是零向量，但 $\sigma^k(\xi)$ 是零向量。证明向量组 $\{\xi,\sigma(\xi), \cdots, \sigma^{k-1}(\xi)\}$ 线性无关。 }

\vspace{0.3cm}

\item  思路：按照线性无关的定义验证。


\end{itemize}

\end{frame}

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\begin{frame}{习题(7.2)\#5.i}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}
\item  {\color{red}问题：设 $\sigma\in L(V)$, 记 $\theta\in L(V)$ 是零变换。
则 $\text{Im}(\sigma)\subseteq \text{Ker}(\sigma)$ 当且仅当 
$$\sigma^2=\theta. $$ } 

\item 证明：
\begin{enumerate}
\item 
\begin{enumerate}
\item 设 $\text{Im}(\sigma)\subseteq \text{Ker}(\sigma)$. 
\item 对任意 $\alpha\in V$, 由变换的幂次的定义可得 $\sigma^2(\alpha)=\sigma(\sigma(\alpha))$. 
\item 由像空间的定义可得 $\sigma(\alpha)\in \text{Im}(\sigma)$, 由条件可得 $\sigma(\alpha)\in \text{Ker}(\sigma)$. 
\item 由核空间的定义可得 $\sigma(\sigma(\alpha))=\theta_V$, 这里 $\theta_V$ 是 $V$ 中的零向量。
\item 因为 $\sigma^2(\alpha)=\theta(\alpha)$ 对任意 $\alpha$ 都成立，所以 $\sigma^2=\theta$. 
\end{enumerate}

\item 
\begin{enumerate}
\item 设 $\sigma^2=\theta$. 
\item 由变换的幂次的定义可得，对任意 $\alpha\in V$, 有 $\sigma(\sigma(\alpha))=\theta(\alpha)=\theta_V$. 
\item 因此根据核空间的定义，对任意 $\alpha\in V$, 有 $\sigma(\alpha)\in\text{Ker}(\sigma)$. 
\item 因此 $\text{Im}(\sigma)\subseteq \text{Ker}(\sigma)$. 
\end{enumerate}

\end{enumerate}

\end{itemize}

\end{frame}

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%\begin{frame}[fragile=singleslide]{3.1.1. }
\begin{frame}{习题(7.2)\#6 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}

\item  {\color{red}问题：记 $\mathbb{R}^n$ 是实数域 $\mathbb{R}$ 上的 $n$ 维行向量空间。定义 $$\sigma(x_1,x_2,\cdots,x_n) = (0,x_1,\cdots,x_{n-1}). $$  } 
\begin{enumerate}
\item  {\color{red}证明 $\sigma$ 是 $\mathbb{R}^n$ 到自身的一个线性变换，且 $\sigma^n=\theta$ 为零变换。} 
\item  {\color{red}求核空间 $\text{Ker}(\sigma)$ 与像空间 $\text{Im}(\sigma)$ 的维数。} 
\end{enumerate}

\vspace{0.3cm}

\item  思路：
\begin{enumerate}
\item  验证线性变换的定义。验证零变换的定义。
\item  按照核空间与像空间的定义，写出这两个子空间的一般元素。
\end{enumerate}

\end{itemize}

\end{frame}

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